The 𝑅_∞-property for nilpotent quotients of Baumslag–Solitar groups

Abstract

A group G has the R∞-property if the number R(ϕ) of twisted conjugacy classes is infinite for any automorphism ϕ of G. For such a group G, the R∞-nilpotency index is the least integer c such that G/γ c+1 (G) still has the R∞-property. In this paper, we determine the R∞-nilpotency degree of all Baumslag-Solitar groups.A central problem is to decide which groups have the R ∞ -property. The study of this problem has been a quite active research topic in recent years. Several families of groups have been studied by many authors. A non-exhaustive list of references is [1,2,3,4,5,6,7,8,9,10,14].Of particular interest for this paper is the fact that in [5] it was proved that the Baumslag-Solitar groups BS(m, n) have the R ∞ -property except for m = n = 1 (or m = n = −1 which is the same group). Recently in [3], motivated by the results of [1], new examples of groups which have the R ∞ -property were obtained by looking at quotients of a group which has the R ∞ -property by the terms of the lower central series as well the derived central series. So it is natural to ask the same question for the groups BS(m, n). Related to this approach, we introduced in [3] the following notion: Definition 1.2. Let G be a group. The R ∞ -nilpotency degree of a group G is the least integer c such that G/γ c+1 (G) has the R ∞ -property. In case this integer does not exist then we say that G has R ∞ -nilpotency degree infinite.In this work we determine the R ∞ -nilpotency degree for all the Baumslag-Solitar groups BS(m, n). The main results of this work are:Theorem 4.5 Let m, n be integers with 0 < m ≤ |n| and gcd(m, n) = 1. Let p denote the largest integer such that 2 p |2m + 2. Then, the R ∞ -nilpotency degree r of BS(m, n) is given by• In case n < 0 and n = −1, then r = 2.• In case n = −1 (so m = 1) then r = ∞.

The $R_\infty$ property for nilpotent quotients of generalized solvable Baumslag-Solitar groups

“…It's also worth remembering that we will restrict us to investigate Reidemeister numbers of Γ n,c only in the case r ≥ 2, for, if r = 1, then Γ n is by definition a Baumslag-Solitar group BS(1, n) and its Reidemeister numbers were studied in [4]. Let ϕ ∈ Aut(Γ n,c ).…”

“…These are important examples of combinatorial and geometric group theory. In [4], K. Dekimpe and D. L. Gonçalves determined the R ∞ -nilpotency degree of BS(m, n): Theorem 2 (Theorem 5.4 in [4]). Let 0 < m ≤ |n| with m = n and take d = gcd(m, n).…”

“…There, one realized that elementary techniques involving the BNS invariant Σ 1 of the groups BS(1, n) could also be applied to Γ n , leading to an elementary proof of R ∞ to these groups. One can then ask whether other techniques -such as the investigation of nilpotent quotients of BS(m, n) in [4] -can also be adapted to Γ n . The main question was whether Γ n would behave more like BS(1, 2) (with infinite R ∞ -nilpotency degree) or like BS(1, n), n ≥ 3 (finite degree cases).…”